Wikipedia:Reference desk/Archives/Science/2020 September 11

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September 11

Time it would take for Earth and Mars to collide

A friend brought up the question of how long it would take for the Earth and Mars to collide. There were several diversions but the core of the question is, if Earth and Mars were the only objects in space, how long would it take for them to collide. We're assuming the gravitational constant remains what it currently is. And no other bodies are affecting them. The distance would be, let's say a light year. And they start out stationary. I thought it would be fairly straightforward. Something along the lines of just using the two masses, gravity, and distance. But I'm not sure how that math problem would be set up. And my Google skills are failing me for what seems like a simple equation. Any help?

I know that school is just starting here in the US but I can assure you this isn't for a school project. I'm a dude in my 40s with no homework due any time soon. Thanks, †dismas†|^(talk) 02:28, 11 September 2020 (UTC)

Per Newton (not Einstein) and his law of universal gravitation

$F=G{\frac {m_{1}m_{2}}{r^{2}}},$

gives the force F pulling two objects together. Solve for

G = 6.674×10−11 m3⋅kg−1⋅s−2., mearth = 5.97237×1024 kg, mmars =  6.4171×1023 kg

Then Newton's 2nd Law of Motion a = F / m

gives the accelerations a_earth and a_mars of each planet towards the other. We can arbitrarily consider one of the planets as a stationary reference. Then the separation distance

s = 1 Light-year - radius_earth - radius_mars

 = 9.46 x 1012 - 6371.0 - 3.39 km

will be closed by the other body accelerating at a = aearth + amars

Remembering the equation of motion s = ut + ½ a t ² the actual calculation of

$t=2sqrt(s/a)$

is left to the reader. 84.209.119.241 (talk) 08:25, 11 September 2020 (UTC)

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I'm afraid that's wrong. Crucially, the gravitational force and hence the acceleration does not stay constant as the two bodies approach each other. I'm not going to attempt to solve the equation of motion (Newton's 2nd law), but merely point to the article free-fall time, which gives the solution. --Wrongfilter (talk) 08:34, 11 September 2020 (UTC)

(ec) To start, I'd make the problem more general. We start out with a Newtonian universe containing two stationary bodies, being point masses of magnitudes $m_{1}$ and $m_{2}$ at a given distance $d_{0}$. Obeying Newton's laws of motion they start moving toward each other at time $t=0$ and their distance $d_{t}$ will decrease as a function of $t$. Solve $d_{t}=0$ for $t$. The centre of mass of the two-body system is stationary at time $t=0$, and Newton's laws (specifically the conservation of momentum) imply it will remain so, so that the bodies will meet there. Likewise, the conservation of angular momentum implies they will move along the straight line connecting them and passing through the centre, which we will take to be the origin. (Even without appeal to conservation of angular momentum, this already follows from elementary considerations of symmetry.) We can therefore think of the two bodies as being located at the positions $d_{1,t}={\tfrac {m_{2}}{m_{1}+m_{2}}}d_{t}$ and $d_{2,t}=-{\tfrac {m_{1}}{m_{1}+m_{2}}}d_{t}$. To simplify the notation, we define $s=s_{t}=d_{1,t}$ and $\mu ={\tfrac {m_{2}}{m_{1}+m_{2}}}$, so that $s=\mu d_{t}$ (and therefore $d=d_{t}=\mu ^{{-}1}s$), and solve $s=0$. The forces attracting the bodies to each other are opposite in sign but equal in magnitude, which is given by $F=Gm_{1}m_{2}d^{{-}2}=$ $G\mu ^{2}m_{1}m_{2}s^{{-}2}=$ $\gamma m_{1}s^{{-}2}$, where $\gamma =G\mu ^{2}m_{2}$.

The acceleration on the first body, taking the proper sign so that the acceleration is towards the origin, is given by ${\ddot {s}}=a=-F/m_{1}=\gamma s^{{-}2}$.

I'll leave it to the two of you to solve this simple ODE and plug in the values for the variables in the model, but don’t be shy to tell us if you need more help. BTW, the planets you want to collide, in their regular orbits, are never more than 4.25 × 10^-5 light-year or 23 light-minutes apart, so putting them a light-year apart is a far stretch; in the universe as we currently understand it, you'd not be able to ignore relativistic effects.  --Lambiam 10:41, 11 September 2020 (UTC)

From the energy conservation law it follows that

${\frac {\mu }{2}}\left({\frac {dr}{dt}}\right)-G{\frac {m_{1}m_{2}}{r}}=-G{\frac {m_{1}m_{2}}{r_{0}}}$,

where $r_{0}$ is the initial distance, $\mu =m_{1}m_{2}/(m_{1}+m_{2})$. Then the time $\tau$ to collision will be

$\tau ={\sqrt {\frac {\mu }{2Gm_{1}m_{2}}}}\int \limits _{r_{E}+r_{M}}^{r_{0}}{\frac {dr}{\sqrt {{\frac {1}{r}}-{\frac {1}{r_{0}}}}}}={\sqrt {\frac {\mu r_{0}^{3}}{2Gm_{1}m_{2}}}}\int \limits _{(r_{E}+r_{M})/r_{0}}^{1}{\frac {{\sqrt {t}}dt}{\sqrt {1-t}}}$.

You can calculate the integral yourself. Ruslik_Zero 20:41, 11 September 2020 (UTC)

I have a faint memory of reading the assertion (possibly due to Newton) that if an orbiting body were to stop suddenly, relative to its primary, it would fall to the barycenter in 1/√32 of its former orbital period. If that's accurate, you need only work out what the period of the Earth-Mars binary would be at that distance, and divide by the appropriate constant. —Tamfang (talk) 00:24, 15 September 2020 (UTC)

It is true and can be derived from the above formula:

$\tau ={\sqrt {\frac {\mu r_{0}^{3}}{2Gm_{1}m_{2}}}}\left({\frac {\pi }{2}}+{\sqrt {a(1-a)}}+\operatorname {asin} {\sqrt {a}}\right)={\sqrt {\frac {\pi ^{2}\mu r_{0}^{3}}{8Gm_{1}m_{2}}}}$,

(where $a=r/r_{0}$) when r=0. Ruslik_Zero 18:47, 16 September 2020 (UTC)

Does milk slosh in a sealed tetrapak?

When I shake a new sealed tetrapak of milk, it always seemed like the milk doesn't slosh. Once I open it and remove the seal it does. Is something about the sealing process preventing the milk from sloshing (even though surely it's still got space to move around)? Or is it because the milk is sloshing, but the sound can't travel through the vacuum? (Which would be very cool, like having a tiny microcosm of space in your milk carton.) --2A01:4C8:64:898F:1:2:A027:D730 (talk) 16:46, 11 September 2020 (UTC)

Milk only sloshes at a phase boundary. If there is no air in the tetrapak, there is no phase boundary against which the milk can slosh. Once opened, air gets in, and now you have the ability to slosh. --Jayron32 17:52, 11 September 2020 (UTC)

So a question that remains, for me, is whether the milk gets mixed when it is shaken, or does one need to open and seal it again before shaking in order to mix the milk?--Shantavira|^{feed me} 18:23, 11 September 2020 (UTC)

If there is no air inside – whether a tetrapack, bottle or can – shaking has no or very little effect. Turning the pack on different sides and slowly keeping turning may help to mix if separated components (e.g. milk and cream) have different densities; this requires no vigorous anything but just patience to let gravity do its thing.  --Lambiam 23:10, 11 September 2020 (UTC)

Tetrapaks aren't used here, so I have no familiarity with opening them, but if there was no air inside, I can't see how there wouldn't be milk spilling out whenever you do. However, there might be just a small bubble of air that could move around without "sloshing". --174.88.168.23 (talk) 01:19, 12 September 2020 (UTC)

Assuming the container isn't rigid, it can slosh even without air by deforming the containiner itself. I just tried a "tall rectangular half-gallon" and shaking it up and down caused it to bulge alternately on the bottom vs top of the sides. That means the liquid is able to move inside even if there is no non-liquid space. DMacks (talk) 20:28, 12 September 2020 (UTC)

An additional consideration is that cow milk froths pretty readily. Depending on how much air is in there (assuming there is some amount), you might not hear much sloshing because the foam is a) deadening the sound and b) filling the previous air gap with a structure of milk foam. Matt Deres (talk) 12:40, 12 September 2020 (UTC)

As to the question about whether the sound can travel through the vacuum, the answer is that it doesn't need to. If you take a Vacuum flask and half fill it with water and then slosh it about, you'll be able to hear the liquid perfectly well despite in that case most of the surrounding space beyond the interior wall being vacuum. The sound waves can pass out by vibrations in the glass of the wall that then get transmitted to the exterior atmosphere. Michael D. Turnbull (talk) 12:52, 13 September 2020 (UTC)

LED Lightbulb Problem

I bought a new LED light bulb for a 3-way lamp, labeled 50-100-150W, which means that its output is equivalent to that of an old 50-100-150W incandescent three-way lightbulb. When I put the bulb into the lamp and cycled the switch, it went 0-50-100-50. I couldn't get it to produce what was supposed to be full illumination at full power. So then I thought that maybe there was something wrong with the lamp, so I switched this new bulb and another existing bulb between two similar lamps. The bulb still switched 0-50-100-50, and did not go up to 150. My question is: What is wrong? It isn't that one of the two sets of light-emitting diodes doesn't work. Each of the sets of diodes works. They just don't work at the same time. Why doesn't the higher-power set of diodes work when I have switched the lamp to energize both sets of diodes? Robert McClenon (talk) 17:50, 11 September 2020 (UTC)

This page talks about incompatibility of using incandescent lamp dimmers for LEDS. It might help, i don't really know much about the topic tbh. Zindor (talk) 18:02, 11 September 2020 (UTC)

Dimmers are irrelevant. The traditional 3-way light is a bulb with two separate filaments powered independently, and does not involve a dimmer. I don't know about Robert's problem but I would suggest contacting the manufacturer. --174.88.168.23 (talk) 01:22, 12 September 2020 (UTC)

Yes, that is a little bit weird. Robert, if you have a replacement incandescent bulb, maybe you can try that to see if the 3-way switch works properly. Also, examine the contacts on your LED bulb to see that they are in the right places. A 3-way LED bulb is kind of a hack, though. LED's are dimmable, though the dimming electronics are different from old-fashioned incandescent (triac) dimmers. Places like homedepot.com will have tons of stuff like that. 2602:24A:DE47:BB20:50DE:F402:42A6:A17D (talk) 02:25, 12 September 2020 (UTC)

I switched the bulb in question with a working 3-way incandescent bulb. The incandescent bulb worked fine. The questionable LED bulb continued to work 0-50-100-50 in another similar lamp. One comment I got is that maybe the complicated electronics in the bulb is trying to control the current to the diode assemblies, and that there is some error in the "intelligence". That is, if you put artificial intelligence in something, you risk putting artificial madness in it. So I have a bulb that doesn't work the way it is supposed to work. Robert McClenon (talk) 04:53, 12 September 2020 (UTC)

"if you put artificial intelligence in something, you risk putting artificial madness in it" sounds like the tagline to a great cyber-dystopia film. Flesh it out a bit with computer screens that cast glowing text on people's faces and maybe some boobs and you're got a hit on your hands. Let me know how to invest in this. Matt Deres (talk) 12:47, 12 September 2020 (UTC)

The phrase should be voiced by Jeff Goldblum, hopefully in a more pithy formulation.  --Lambiam 20:26, 12 September 2020 (UTC)

Though this be artificial madness, yet there is an algorithm in 't.  --Lambiam 20:30, 12 September 2020 (UTC)

It's worth closely reading the packaging to check whether it actually promises that it lights all the LEDs with both live contacts powered, or if it just promises to be "compatible" with 3-way lamp fixtures. Every standard bulb is compatible with a 3-way fixture because the fixture was designed to be "backwards-compatible"; there's just a second hot contact for the second filament in an incandescent. In a "3-way LED lamp", the power controller has to monitor both hot contacts and switch on all the LEDs when they're both powered. A chintzy bulb that was designed to be super-cheap while fooling you could shave off components. (Electronics people: would the bulb work as described if it just connected both hots to a common input? That would explain why it doesn't have a "150W" mode.) --47.146.63.87 (talk) 23:25, 12 September 2020 (UTC)

Robert, what is the brand and model of your particular bulb? -- ToE 16:08, 13 September 2020 (UTC)

User:Thinking of England - General Electric Soft-White 3-way LED bulb. Robert McClenon (talk) 05:10, 14 September 2020 (UTC)

I still suggest contacting the manufacturer. --174.88.168.23 (talk) 05:51, 16 September 2020 (UTC)

Reduction potential at mercury electrode

Hydrogen has high over-potential at mercury electrode. But I could not find the value when I searched Google. What are the reduction potentials of hydrogen, sodium, water and oxygen at mercury electrode in alkaline media? Thanks. Horus1927 20:21, 11 September 2020 (UTC) — Preceding unsigned comment added by Horus1927 (talk • contribs)

Many are tabulated at Standard electrode potential (data page). Usually, hydrogen is set at zero and any other is measured relative to that. Michael D. Turnbull (talk) 15:43, 13 September 2020 (UTC)